I am trying to find out how to calculate the number of audio samples in a note length, when a sample rate and tempo is provided.

For example a 1/4th note at 120bpm 44.1kHz = 22050 samples

From knowing this how would I then work out an 1/8th note?

Really?

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"For example a 1/4th note at 120bpm 44.1kHz = 22050 samples"

not if we´re in a 6/8 signature.

-110

In music software bpm usually refers to quarter notes regardless of signature.

By the way, to answer the question, (grin)

Divide by 2!

I’m much more interested to understand the purpose of the calculation.

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